The Fifty-Second William Lowell Putnam Mathematical Competition Saturday, December 7, 1991 ✄ ✂✁☎✄ ✄ rectangle has vertices as ✆✞✝✠✟✡✝☞☛✌✟✍✆ ✟✎✝✏☛✑✟✒✆✓✝✔✟ ☛✌✟ and ✆ ✟ ☛ It rotates ✕✏✝✏✖ clockwise about the point ✆ ✟✡✝☞☛ It then rotates ✕✗✝☞✖ clockwise about the point ✆✙✘✚✟✡✝☞☛ , then ✕✗✝✏✖ clockwise about the point ✆✜✛✢✟✡✝☞☛ , and finally, ✕✏✝✏✖ clockwise about the point ✆✤✣✍✝✠✟✎✝✏☛ (The side originally on the ✥ -axis is now back on the ✥ -axis.) Find the area of the region above the ✥ -axis and below the curve each ✈ ❇ is in the sequence ✣✗✟ ✟✡②✔✟✰❍✒❍✰❍③✟✤④✾⑤✗✟✒❍✰❍✒❍ de fined by ④☞✽⑥✫●✣ , ④ ✯ ✫ , and ④✾⑤r✫❉④✾⑤ ✽✍✴r④✾⑤ ✯ ✴✳✣✗✟ t t and if ✈✰✽⑦✫❲④✏⑧ then every element in ⑨✏✣✗✟ ✟✡②✔✟✰❍✒❍✰❍③✟✤④✏⑧✢⑩ appears at least once as a ✈ ❇ A–1 A traced out by the point whose initial position is (1,1) ✁ A–2 Let ✦ and ✧ be different ★ ★ matrices with real entries If ✦✪✩✬✫✭✧✮✩ and ✦✪✯✰✧✱✫✲✧✳✯✰✦ , can ✦✪✯✵✴✶✧✳✯ be invertible? A–3 Find all real polynomials ✷✸✆✓✥✹☛ of degree ★✻✺ for which there exist real numbers ✼✾✽❀✿❁✼ ✯ ✿❃❂✒❂✰❂❄✿❅✼✒❆ such that ✷✸✆✓✼✒❇✜☛❈✫❉✝✔✟ ❊❋✫●✣✏✟ ✒✟ ❍✰❍✒❍✑✟■★❏✟ and ✫❲✝ ❊❳✫●✣✏✟ ✟✰❍✰❍✒❍✌✟■★❩❨❬✣✗✟ ✷▲❑◆▼✹❖✤❘◗✂❖■❚❙✢❯ ✯ ❱ where ✷ ❑ ✆✓✥✹☛ denotes the derivative of ✷❳✆❘✥✹☛ A–4 Does there exist an infinite sequence of closed discs ❭❪✽✾✟✎❭ ✟✡❭ ✟✒❍✰❍✒❍ in the plane, with centers ✩ ❫ ✽✾✟ ❫ ✟ ❫ ✟✒❍✰❍✰❍ ,✯ respectively, such that ✯ ✩ the ❫ ❇ have no limit point in the finite plane, the sum of the areas of the ❭❴❇ is finite, and every line in the plane intersects at least one of the ❭ ❇? A–5 Find the maximum value of ❵✶❛ ❜❞❝ ✥▲❡❢✴❣✆✓❤✐❨❀❤ ✯ ☛ ✯❦❥ ✥ Prove that ♦✮✆❘★❦☛❶✫❲✉✪✆✓★❦☛ for each ★❀✺❲✣ ♦✐✆✜✛❷☛❸✄ ✫❹✘ because the relevant sums are ✛✢✟✎❺❢✴❻✣✗✟③✘☎✴ ✟■②❢✴ ✟■②✵✴ ✴❻✣✏✟ and ✉♠✆✙✛❷ ☛❶✫♥✘ because the relevant sums are ②❄✴ ✴❬✣✏✟ ✴ ✴ ✴❼✣✗✟ ✴ ✴ ✣▲✴❽✣▲✴❾✣✗✟ ✴❽✣▲✴❾✣▲✴❾✣▲✴❽✣✔✴❾✣❷✟✰✣✔✴❾✣✠✴❾✣✔✴❽✣✔✴❾✣✔✴❽✣✔✴❾✣❷❍ ) (For example, ★❼✺✭✝ , ♣ let ❿⑥✆✓★❦☛❄♣ ✫❹★❾❨➁♣ ➀➂✯ , where ➀ B–1 For ♣ each integer ♣ is the greatest integer with ➀❩✯➃❧❉★ Define a sequence ✆ ⑧❷☛✡➄⑧✌➅ ❜ by ❜ ✫❲♦ and ⑧ ✽❄✫ ⑧☎✴➆❿⑥✆ ⑧❷☛ for ➇♠✺❬✝ ◗ For what positive integers ♦ is this sequence eventually constant? B–2 Suppose ➈ and ④ are non-constant, differentiable, realvalued functions defined on ✆✤❨r➉❣✟③➉❻☛ Furthermore, suppose that for each pair of real numbers ✥ and ❤ , ➈❋✆❘✥✮✴➆❤✠☛❻✫➊➈❋✆✓✥✹☛■➈❋✆✓❤✠☛❳❨➋④s✆❘✥◆☛✜④s✆❘❤✠☛✌✟ ④s✆❘✥✮✴➆❤✠☛❻✫➊➈❋✆✓✥✹☛✜④s✆❘❤✠☛✂✴➁④◆✆✓✥✹☛✡➈❋✆❘❤✠☛✌❍ If ➈ ✥ ❑ ✆✞✝✏☛❸✫●✝ , prove that ✆✙➈❋✆❘✥✹☛✡☛✤✯❸✴✲✆➌④s✆❘✥◆☛■☛■✯✮✫➍✣ for all B–3 Does there exist a real number ➎ such that, ✁ if ➀ and ★ are integers greater than ➎ , then ✁an ➀ ★ rectangle ✁ ❺ and ✘ ✛ rectmay be expressed as a union of ② angles, any two of which intersect at most along their boundaries? B–4 Suppose ✷ is an odd prime Prove that ➐ for ✝❴❧✶❤♠❧♥✣ A–6 Let ♦✮✆❘★❦☛ denote the number of sums of positive inte♣ ♣ ♣ gers ✽❏✴ ✴ ❂✰❂✰❂✾✴ ✯ ❣ ❖ ♣ ♣ ♣ ♣ which add ♣ up to ♣ ★ with ♣ ♣ ♣ ♣ ♣ ✽rq ✯ ✴ ✩ ✟ ✯ q ✩ ✴ ❡ ✟✰❍✒✰❍ ❍s✟ q ✽❏✴ ✟ ✽❄q ❍ ❖✑t ✯ ❖✰t ❖ ❖✰t ❖ Let ✉♠✆❘★❦☛ denote the number of ✈✰✽✔✴❩✈ ✴✇❂✰❂✒✜❂ ✴❩✈✑① which ✯ add up to ★ , with ✈✰✽r✺❉✈ ✯ ❲ ✺ ❂✒❂✰❂✠✺❉✈✑①✍✟ ➏ ⑤✡➅ ❜➃➑ ✷➒☞➓ ➑ ➒ ✷✳✴ ➒➊➓❲➔ ➏ ✴❣✣→✆❘➣❴↔✚↕✳✷ ✯ ☛✑❍ B–5 Let ✷ be an odd prime and let ➙ ➏ denote (the field of) integers modulo ✷ How many elements are in the set ⑨✒✥ ✯➜➛ ✥❾➝❩➙ ➏ ⑩☎➞➋⑨✍❤ ✯ ✴❣✣ ➛ ❤✪➝❩➙ ➏ ⑩✾➟ ♣ B–6 Let and ✈ be positive numbers Find the largest number ❫ , in♣✢terms ➠ ➠ of ♣❋and ➡■➢➥➤✠➦⑥✈ ,➧ such that ➡■➢➥➤✠➦⑥➧ ➡■➢➥➤✠➦⑥➧ ✥ ➡✡➢❚➤✠✆✤➦⑥✣✵➧ ❨✇✥◆☛ ✈✽t ❧ ✴✶✈ ➧ ➧ ➦⑥➧ ✝➨✿➫➩ ➩❏❧ ❫ and for all ✥ , ✝➁✿➭✥❲✿❅✣ for all ➡✡➢➥➤✠with (Note: ✫●✆✞➯✾➲➃❨➨➯ t ➲☞☛✎➳ ) ♣